Formation of singularities in one-dimensional Chaplygin gas
aa r X i v : . [ m a t h . A P ] N ov FORMATION OF SINGULARITIES IN ONE-DIMENSIONALCHAPLYGIN GAS
De-Xing Kong a Changhua Wei b, ∗ Qiang Zhang c ∗ Corresponding author a Department of Mathematics, Zhejiang University, Hangzhou 310027, ChinaE-mail address: [email protected] b Department of Mathematics, Zhejiang University, Hangzhou 310027, ChinaE-mail address: wch [email protected] c Department of Mathematics, City University of Hong Kong, ChinaE-mail address: [email protected]
Abstract.
In this paper we investigate the formation and propagation of singularitiesfor the system for one-dimensional Chaplygin gas. In particular, under suitable assump-tions we construct a physical solution with a new type of singularities called “Delta-like”solution for this kind of quasilinear hyperbolic system with linearly degenerate charac-teristics. By careful analysis, we study the behavior of the solution in a neighborhoodof a blowup point. The formation of this new kind of singularities is due to the envelopeof the different families of characteristics instead of the same family of characteristics inthe traditional situation. This shows that the blowup phenomenon of solution for thesystem with linearly degenerate characteristics is quite different from the situation ofshock formation for the system with genuinely nonlinear characteristics. Different initialdata can lead to kinds of different Delta-like singularities: the Delta-like singularity withpoint-shape and Delta-like singularity with line-shape.
Key words and phrases:
System for Chaplygin gas, linearly degenerate characteristic,blowup, singularity, Delta-like singularity. : 35L45, 35L67, 76N15. Introduction
As we know, smooth solutions of nonlinear hyperbolic systems generally exist in finitetime even if initial data is sufficiently smooth and small. After this time, only weaksolutions can be defined. Therefore, the following questions arise naturally:(I) When and where do the solutions blow up?(II) What quantities blow up? how do they blow up?(III) What kinds of singularities appear? How do the singularities propagate?These questions are very important in mathematics and physics. For questions (I) and(II), some methods were established and many results were obtained (see [9], [12], [17],[18]). As for question (III), since this kind of nonlinear phenomena is too complicated, up o now, only a few results on shock formation are known. For a single conservation law,these questions can be solved well by the usual characteristic method (see [17]). For the p -system, Lebaud [15] investigates the problem of shock formation from the simple waves,namely, under the hypothesis that one Riemann invariant keeps constant. Kong [10]studies the formation and propagation of singularities (in particular, the shock formation)for 2 × lowup point, the density function of the gas is infinite, this phenomenon is due to theconcentration of mass of the gas in finite interval.The rest of the paper is organized as follows. In Section 2, we give some preliminar-ies on quasilinear hyperbolic system with linearly degenerate characteristics. In Section3, using the method of characteristic coordinates and the singularity theory of smoothmappings, we give a detailed analysis on the formation of singularities. In Section 4, wepresent a complete description of the solution in the neighborhood of the blowup point.In Section 5, we construct a physical solution containing a new kind of singularity called“Delta-like” singularity after the blowup time. In Section 6, under different assumptionson initial data, we construct several different weak solutions with “Delta-like” singularitieswhich are named the “Delta-like singularity with point-shape” and “Delta-like singularitywith line-shape”, respectively. Section 7 is for conclusion.2. Preliminaries
In this section, we consider one-dimensional system of isentropic gas in Eulerian rep-resentation ( ∂ t ρ + ∂ x ( ρu ) = 0 ,∂ t ( ρu ) + ∂ x ( ρu + p ) = 0 , (2 . t ∈ R + and x ∈ R stand for the time variable and spatial variable, respectively,while ρ = ρ ( t, x ) and u = u ( t, x ) denote the density and the velocity, respectively, and p ( t, x ) is the pressure which is a function of ρ given by p = p − µ ρ . (2 . p and µ are two positive constants. The system (2.1) with (2.2) describes the motionof a perfect fluid characterized by the pressure-density relation (known as the Chaplygin orvon K´arm´an-Tsien pressure law). This endows the system a highly symmetric structure.This is evident if we adopt the local sound speed c = c ( t, x ) = [ p ′ ( ρ )] / and the usualmean velocity of the fluid u as dependent variables. In this case, the system reads ∂ t U + A ( U ) ∂ x U = 0 , (2 . U = (cid:18) cu (cid:19) and A(U) = (cid:18) u − c − c u (cid:19) . bviously, the eigenvalues of A ( U ) read λ − = u − c, λ + = u + c. (2 . λ ± = λ ± ( t, x ) are Riemann invariants. Under theRiemann invariants, the system (2.1) can be reduced to ( ∂ t λ − + λ + ∂ x λ − = 0 ,∂ t λ + + λ − ∂ x λ + = 0 . (2 . t = 0 : ρ = ρ ( x ) , u = u ( x ) , (2 . ρ ( x ) and u ( x ) are two suitably smooth functions with bounded C norm. Forconsistency, let λ ± (0 , x ) = Λ ± ( x ) , u ( x ) ± µρ ( x ) . (2 . , α ) by x = x + ( t, α ) , x = x − ( t, α ) , respectively, which satisfy dx + ( t, α ) dt = λ − ( t, x + ( t, α )) ,t = 0 : x + (0 , α ) = α (2 . dx − ( t, α ) dt = λ + ( t, x − ( t, α )) ,t = 0 : x − (0 , α ) = α, (2 . α, β ) be the characteristic parameters defined as follows. For any( t, x ) in the maximal domain of definition of a smooth solution , we define β ( t, x ) by β ( t, x ) = x + (0; t, x ) where x + (0; t, x ) is the unique solution of the ODE df ( s ; t, x ) ds = λ − ( s, f ( s ; t, x )) , with initial condition f ( t ) = x . α ( t, x ) is defined similarly. The geometric meaning of α ( t, x ) and β ( t, x ) are shown in Figure 1. emma 2.1. By (2.5), λ + ( t, x ) is constant along the curve x = x + ( t, α ) , while λ − ( t, x ) is constant along the curve x = x − ( t, α ) . The following lemma can be found in Kong-Zhang [14]
Lemma 2.2.
In terms of characteristic parameters ( α, β ) introduced above, it holds that t ( α, β ) = Z βα + ( ζ ) − Λ − ( ζ ) dζ , (2 . x ( α, β ) = 12 (cid:26) α + β + Z βα Λ + ( ζ ) + Λ − ( ζ )Λ + ( ζ ) − Λ − ( ζ ) dζ (cid:27) . (2 . ✲✻ t xα ( t, x ) β ( t, x )0 ( t, x ) x − x + Figure 1.
The geometric meaning of characteristic coordinates α and β .3. formation of singularity This section is devoted to the formation of the cusp-type singularity under suitableassumptions on initial data. The following lemma, which can be found in Kong [13], playsan important role in our discussion.
Lemma 3.1.
Adopt the notations in Section 2. If there exists α such that Λ − ( α ) = λ + ( t, x − ( t, α )) for t ≥ , then it holds that ∂x − ( t, α ) ∂α = λ + ( t, x − ( t, α )) − Λ − ( α )Λ + ( α ) − Λ − ( α ) (3 . and ∂λ − ( t, x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = x − ( t,α ) = Λ ′ − ( α ) Λ + ( α ) − Λ − ( α ) λ + ( t, x − ( t, α )) − Λ − ( α ) . (3 . Similar result holds for x = x + ( t, β ) and λ + ( t, x + ( t, β )) . emark 3.1. It follows from (3 . that if there exists time t > which satisfies λ + ( t , x − ( t , α )) = Λ − ( α ) , Λ − ( α ) = Λ + ( α ) and Λ ′ − ( α ) = 0 , for some α, then the solution of the Cauchy problem (2.5), (2.7) must blow up at the time t . By thetheory of characteristic method, we observe that λ − ( t, x ) and λ + ( t, x ) are bounded, while ( λ − ) x and ( λ + ) x tend to the infinity as t goes to t .It is well known that the formation of traditional blowup, e.g., the formation of “shockwave” is due to the envelope of the same family of characteristics (see [1, 12] ). However,in this paper, we shall investigate a new phenomenon on the formation of singularitieswhich is based on the envelope of different families of characteristics (see Figure 2). ✲✻ t x α β ( t, x ) x − x + Figure 2.
The envelope of different families of characteristicsTo do so, we suppose that the initial data Λ − ( x ) and Λ + ( x ) are suitably smoothfunctions and satisfy the following assumptions:Assumption (H1): Λ − ( x ) < Λ + ( x ) , ∀ x ∈ R . (3 . ′ − ( x ) < ′ + ( x ) < , ∀ x ∈ R . (3 . { ( α, β ) | α < β and Λ − ( α ) = Λ + ( β ) } . (3 . − ( x ) by α andthe variable of Λ + ( x ) by β .By (3.4) and (3.5), for ∀ ( α, β ) ∈ Σ, it holds that β ( α ) = Λ − Λ − ( α ). Define f ( α ) , Λ ′ − ( α )Λ + ( β ( α )) − Λ − ( β ( α )) − Λ ′ + ( β ( α ))Λ + ( α ) − Λ − ( α ) , (3 . here Λ ′ + ( β ( α )) = d Λ + ( β ) dβ (cid:12)(cid:12)(cid:12)(cid:12) β = β ( α ) . We furthermore assume that there exists ( α , β ) ∈ Σ such thatAssumption (H3): Λ − ( α ) = Λ + ( β ) . (3 . f ( α ) = 0 . (3 . f ′ ( α ) < . (3 . − ( α ) = Λ + ( β ) = 0 . (3 . Lemma 3.2.
Initial data set { (Λ + ( x ) , Λ − ( x )) } satisfying assumptions (H1)-(H5) is notempty.Proof. We prove the lemma by construction.Firstly, choose Λ − ( x ) such that it satisfies (3.4) and (3.10). Then, at the point α itholds that Λ − ( α ) = 0 and Λ ′ − ( α ) < . Secondly, fix Λ − ( x ) and choose Λ + ( x ) > Λ − ( x ) for all x ∈ R . Moreover, at the point β it satisfies (3.8) and Λ + ( β ) = 0.By (3.8), Λ + ( x ) satisfies Λ ′ − ( α )Λ + ( β ) − Λ − ( β ) = Λ ′ + ( β )Λ + ( α ) − Λ − ( α ) , By assumption (H1), it holds thatΛ + ( β ) − Λ − ( β ) > + ( α ) − Λ − ( α ) > , and then Λ ′ + ( β ) < . Thus, we can choose Λ + ( x ) satisfing assumptions (H2)-(H4).Finally, we prove that Λ + ( x ), constructed in the way mentioned above, satisfies theassumption (H5) for fixed Λ − ( x ). n fact, by (3.8), if we fix the value of Λ ′ + ( β ) <
0, then Λ + ( α ) satisfiesΛ + ( α ) = − Λ ′ + ( β )Λ − ( β )Λ ′ − ( α ) > . By (3.5) and (3.9), it must hold that f ′ ( α ) = − Λ ′′ − ( α )Λ − ( β ) + Λ ′ − ( α ) (cid:0) Λ ′ + ( β ) − Λ ′ − ( β ) (cid:1) Λ ′− ( α )Λ ′ + ( β ) Λ − ( β ) − Λ ′′ + ( β ) Λ ′− ( α )Λ ′ + ( β ) Λ + ( α ) − Λ ′ + ( β ) (cid:0) Λ ′ + ( α ) − Λ ′ − ( α ) (cid:1) Λ ( α )= Λ ′′ − ( α )Λ ( α ) − Λ − ( β )Λ ′′ + ( β ) − Λ − ( β )Λ ( α ) + Λ ′ + ( β )Λ − ( β ) (cid:2) Λ ′ + ( β ) − Λ ′ − ( β ) − (cid:0) Λ ′ + ( α ) − Λ ′ − ( α ) (cid:1)(cid:3) − Λ − ( β )Λ ( α ) < , (3.11)where β = β ( α ). Since Λ ′ − ( x ) <
0, by (3.4) and (3.5), we haveΛ − ( β ) < . Then, at β = β ( α ), Λ + ( x ) should satisfy at α and β Λ − ( β )Λ ′′ + ( β ) > Λ ′′ − ( α )Λ ( α )+Λ ′ + ( β )Λ − ( β ) h Λ ′ + ( β ) − Λ ′ − ( β ) − (cid:16) Λ ′ + ( α ) − Λ ′ − ( α ) (cid:17)i . Obviously, there exists such a Λ + ( x ) such that the above inequality holds at the point α and β = β ( α ). Therefore, it is easy to construct a smooth curve Λ + ( x ) to satisfyassumptions (H1)-(H5) once the information has been known at the points α and β . (cid:3) Remark 3.2.
Assumptions (H3)-(H5) are restrictions to the initial data Λ ± ( x ) at thepoints α and β , so we can change the shape of the curves Λ ± ( x ) to make sure that theysatisfy assumptions (H1)-(H5) once their properties at P = ( α , and P = ( β , havebeen known. In fact, the geometric meaning of assumption (H4) is | BA | = | CD | as shown in Figure 3, where | · | denotes the distance in Euclidean space, P B (resp. P D )stands for the tangential line of the curve Λ − ( x ) (resp. Λ + ( x ) ) at the point P (resp. P ). By the existence and uniqueness theorem of a C solution of the Cauchy problem fora quasilinear hyperbolic systems (see [8]), under the assumptions (H1)-(H5), the Cauchyproblem (2.5)-(2.7) has a unique C solution ( λ − ( t, x ) , λ + ( t, x )) in the domain D ( t ) , { ( t, x ) | ≤ t < t , −∞ < x < ∞} , where t is just the blowup time, i.e., the life span of BA D C xP P Λ − ( x ) Λ + ( x ) Figure 3.
The geometric meaning of the assumption (H4).the C solution of the Cauchy problem (2.5)-(2.7). Throughout the paper, we refer D ( t )as the existence domain of the classical solution.The next lemma comes from Kong [13]. Lemma 3.3.
If there exist two points α and β satisfying (3.7), then the characteristic x = x − ( t, α ) must intersect x = x + ( t, β ) in finite time, where we assume that theclassical solution exists. In what follows, under the assumptions (H1)-(H5), we consider the Cauchy problemgiven by (2.5), (2.7).Let us fix ( α , β ) satisfying the assumptions (H1)-(H5), we introduce t = Z β α + ( ζ ) − Λ − ( ζ ) dζ (3 . x = 12 (cid:26) α + β + Z β α Λ + ( ζ ) + Λ − ( ζ )Λ + ( ζ ) − Λ − ( ζ ) dζ (cid:27) . (3 . Lemma 3.4.
There exists a positive constant ǫ such that α is the unique zero point of f ( α ) , i.e., f ( α ) = 0 but f( α ) = 0 , for α ∈ ( α − ǫ, α + ǫ ) . Proof.
The result comes from (3.8) and (3.9) directly. (cid:3)
It is obvious that (2.10)-(2.11) define a mapping from the region U , { ( α, β ) | α ≤ β } to the domain { ( t, x ) | t ≥ , x ∈ R } . Denote it by Π :Π( α, β ) = ( t ( α, β ) , x ( α, β )) . (3 . △ ( α, β ) = (cid:18) t α t β x α x β (cid:19) (3 . nd its Jacobian J ( α, β ) = t α x β − t β x α . (3 . Definition 3.1.
A point p in U is called a regular point of the mapping Π if the rank △ is 2 at p . Otherwise, p is called a singular point of Π . It is easy to verify that p is a singular point is equivalent to Λ − ( α ) = Λ + ( β ), whichcan form a smooth curve defined by an explicit function β = β ( α ), since Λ ′ + ( β ) < Definition 3.2.
Let p be a singular point of Π and Υ( α ) = ( α, β ( α )) be the parametricequation with Υ( α ) = p for J ( α, β ) = 0 . p is called a fold point of Π , if ddα (Π ◦ Υ)( α ) =(0 , , and p is called a cusp point of Π , if ddα (Π ◦ Υ)( α ) = (0 , but d dα (Π ◦ Υ)( α ) = (0 , . Lemma 3.5. (A) The curve β = β ( α ) is strictly increasing as a function of α ; (B) thesingular points ( α, β ) = ( α , β ) are fold points, while ( α , β ) is a cusp point.Proof. Differenting Λ − ( α ) = Λ + ( β ) with respect to α givesΛ ′ − ( α ) = Λ ′ + ( β ) β α , (3 . β α = Λ ′ − ( α )Λ ′ + ( β ) > . (3 . β = β ( α ) is strictly increasing as a function of α .This proves Part (A).We next prove Part (B).To do so, we notice that along β = β ( α ) ddα ( t ( α, β ) , x ( α, β ))= ddα (cid:18)Z βα + ( ζ ) − Λ − ( ζ ) dζ , (cid:26) α + β + Z βα Λ + ( ζ ) + Λ − ( ζ )Λ + ( ζ ) − Λ − ( ζ ) dζ (cid:27)(cid:19) = (cid:18) β α Λ + ( β ) − Λ − ( β ) − + ( α ) − Λ − ( α ) , Λ + ( β ) β α Λ + ( β ) − Λ − ( β ) − Λ − ( α )Λ + ( α ) − Λ − ( α ) (cid:19) = Λ ′ − ( α )(Λ + ( β ) − Λ − ( β ))Λ ′ + ( β ) − + ( α ) − Λ − ( α ) , Λ + ( β )Λ ′ − ( α )(Λ + ( β ) − Λ − ( β ))Λ ′ + ( β ) − Λ + ( β )Λ + ( α ) − Λ − ( α ) ! = 1Λ ′ + ( β ) Λ ′ − ( α )Λ + ( β ) − Λ − ( β ) − Λ ′ + ( β )Λ + ( α ) − Λ − ( α ) , Λ + ( β ) ( Λ ′ − ( α )Λ + ( β ) − Λ − ( β ) − Λ ′ + ( β )Λ + ( α ) − Λ − ( α ) )! . he assumptions (H4)-(H5) yield ddα ( t ( α, β ) , x ( α, β )) (cid:12)(cid:12) ( α,β ) =( α ,β ) = 0 . On the other hand, along the curve β = β ( α ) d dα ( t ( α, β ) , x ( α, β ))= ddα (cid:18) ′ + ( β ) f ( α ) , Λ + ( β )Λ ′ + ( β ) f ( α ) (cid:19) = − Λ ′′ + ( β ) β α (Λ ′ + ( β )) f ( α ) + 1Λ ′ + ( β ) f ′ ( α ) , [(Λ ′ + ( β )) − Λ + ( β )Λ ′′ + ( β )] β α (Λ ′ + ( β )) f ( α ) + Λ + ( β )Λ ′ + ( β ) f ′ ( α ) ! . The assumptions (H4)-(H5) again give ddα ( t ( α, β ) , x ( α, β )) (cid:12)(cid:12) ( α,β )=( α ,β ) = (0 , , but d dα ( t ( α, β ) , x ( α, β )) (cid:12)(cid:12) ( α,β )=( α ,β ) = (0 , . Thus, the singular points ( α, β ) = ( α , β ) are fold points, while ( α , β ) is a cusp point. (cid:3) Lemma 3.6.
Under the assumptions (H1)-(H5), t is the unique minimum point on theinterval ( α − ǫ, α + ǫ ) , where ǫ is given in Lemma 3.4.Proof. From Lemma 3.4 and by a straightforward calculation we have dtdα (cid:12)(cid:12)(cid:12)(cid:12) α = α = 0 , d tdα (cid:12)(cid:12)(cid:12)(cid:12) α = α > . (3 . (cid:3) We next discuss the position and property of Π( α, β ( α )) in the ( t, x )-plane.Introduce Υ l as the graph of the curve β = β ( α ) with domain ( α − ǫ, α ) and Υ r as the graph of the curve β = β ( α ) with domain ( α , α + ǫ ). Then we define Γ l =Π(Υ l ) and Γ r = Π(Υ r ). We have the following lemma. Lemma 3.7.
Under the assumptions (H1)-(H5), Γ l and Γ r form a smooth curve in (t,x)-plane which can be defined by an explicit function t = t ( x ) , moreover, Γ l is increasing andconcave with respect to x , Γ r is decreasing and concave with respect to x .Proof. By Lemma 3.5, we have dtdα = f ( α )Λ ′ + ( β ) , dxdα = Λ + ( β )Λ ′ + ( β ) f ( α ) . (3 . ccording to (3.8) and (3.9), f ( α ) > , Λ + ( β ) > , ∀ α ∈ ( α − ǫ, α )and f ( α ) < , Λ + ( β ) < , ∀ α ∈ ( α , α + ǫ ) . So by (3.21) and (3.4), dtdα = f ( α )Λ ′ + ( β ) < , dxdα = Λ + ( β )Λ ′ + ( β ) f ( α ) < , ∀ α ∈ ( α − ǫ, α ) . By the implicit function theorem Γ l form a smooth curve t = t ( x ). Moreover dtdx > , so Γ l is increasing with respect to x .On the other hand, for α ∈ ( α , α + ǫ ) it holds that dtdα = f ( α )Λ ′ + ( β ) > , dxdα = Λ + ( β )Λ ′ + ( β ) f ( α ) < . This gives dtdx < . Thus, Γ r is decreasing with respect to x .Moreover, since d tdx = ddx ( dtdx ) = ddx ( 1Λ − ( α ) ) = ddα ( 1Λ − ( α ) ) dαdx = − Λ ′ − ( α )Λ ′ + ( β )Λ − ( α )Λ + ( β ) f ( α ) < , we have d tdx < , ∀ α ∈ ( α − ǫ, α ) . This implies that Γ l is concave with respect to x . Similarly, we have d tdx < , ∀ α ∈ ( α , α + ǫ ) , namely, Γ r is concave with respect to x . (cid:3) Based on the properties derived in Lemmas 3.5-3.7, we can sketch the map from ( α, β )to ( t, x ) (see Figure 4).
Remark 3.3.
From ( t , x ) , there exist only two characteristics which intersect the x -axisat α and β , respectively. Then, at ( t , x ) it holds that dxdt = Λ − ( α ) = Λ + ( β ) = 0 , namely, the two characteristics are tangent at ( t , x ) . ✻ ✲ ✲✻ β α α α α Π t x t , x )Γ r Γ l Figure 4.
The mapping Π under the assumptions (H1)-(H5).Lemmas 2.1, 3.1, 3.5 and 3.6 lead to the following main result.
Theorem 3.1.
Under the assumptions (H1)-(H5), the smooth solution of Cauchy problem(2.5) and (2.7) blows up at ( t , x ) , which is defined by (3.12)-(3.13), and t is the blowuptime. Furthermore, the blowup is geometric blowup. Remark 3.4.
The geometric blowup comes from Alinhac [1] . Roughly speaking, the solu-tion itself keeps bounded, however, the derivatives of first order go to infinity when ( t, x ) tends to the blowup point. Remark 3.5.
In the domain bounded by Γ l and Γ r , characteristics of the same familymust intersect, see Figure 5. ✲✻ t x α α α β β β x − x + Γ r Γ l Figure 5.
The characteristics4.
Estimates of singularities
In this section we shall establish some estimates for the solution near the blowuppoint, these estimates describe the behavior of singularities near the blowup point. Inwhat follows, we focus on the domain O ǫ = { ( t, x ) | ( t − t ) + ( x − x ) < ǫ } . Let e t = t − t , e x = x − x , e α = α − α , e β = β − β . emark 4.1. Throughout the paper, without special notations, the above symbols areadopted to denote the differences of the vector components between regular points and theblowup point.
We have the following theorem
Theorem 4.1.
Under the assumptions (H1)-(H5), in the neighborhood of the blowuppoint, it holds that for any ( t, x ) ∈ O ǫ \ ( t , x ) if e x = o ( | e t | ) , | u ( t, x ) − u ( t , x ) | ≤ F | e t | + F | e x e t | , | ρ | ≤ F | e t | − , | u x | ≤ F | e t | − , | u t | ≤ F + F | e x || e t | , | ρ x | ≤ F | e t | − , | ρ t | ≤ F | e t | + F | e x || e t | , if | e t | = o ( e x ) , | u ( t, x ) − u ( t , x ) | ≤ F | e x | , | ρ | ≤ F | e x | − , | u x | ≤ F | e x | − , | u t | ≤ F | e x | − , | ρ x | ≤ F | e x | − , | ρ t | ≤ F | e x | − , if e x = O (1) | e t | , | u ( t, x ) − u ( t , x ) | ≤ F | e x | , | ρ | ≤ F | e t | − , | u x | ≤ F | e t | − , | u t | ≤ F | e t | − , | ρ x | ≤ F | e t | − , | ρ t | ≤ F | e t | − , for sufficiently small e t and e x , where F i ( i = 1 , , · · · , are positive constants dependonly on the initial data at ( α , β ) and the symbol O (1) denote a quantity whose absolutevalue is bounded depending on the relationship between e x and | e t | when e x is sufficientlysmall. t follows from (2.10), (2.11), (3.12) and (3.13) that e x = 12 (cid:26) α − α + β − β + Z βα Λ + ( ζ ) + Λ − ( ζ )Λ + ( ζ ) − Λ − ( ζ ) dζ − Z β α Λ + ( ζ ) + Λ − ( ζ )Λ + ( ζ ) − Λ − ( ζ ) dζ (cid:27) = Z αα − Λ − ( ζ )Λ + ( ζ ) − Λ − ( ζ ) dζ + Z ββ Λ + ( ζ )Λ + ( ζ ) − Λ − ( ζ ) dζ = Λ + ( β )Λ + ( β ) − Λ − ( β ) e β − Λ − ( α ) e α Λ + ( α ) − Λ − ( α )+ 12 Λ ′ + ( β )Λ + ( β ) − Λ − ( β ) − Λ + ( β )(Λ ′ + ( β ) − Λ ′ − ( β ))(Λ + ( β ) − Λ − ( β )) ! e β − Λ ′ − ( α )Λ + ( α ) − Λ − ( α ) − (Λ ′ + ( α ) − Λ ′ − ( α ))Λ − ( α )(Λ + ( α ) − Λ − ( α )) ! e α + 16 " Λ ′′ + ( β )Λ + ( β ) − Λ − ( β ) − Λ ′ + ( β ) (cid:0) Λ ′ + ( β ) − Λ ′ − ( β ) (cid:1) (Λ + ( β ) − Λ − ( β )) − Λ ′ + ( β ) (cid:0) Λ ′ + ( β ) − Λ ′ − ( β ) (cid:1) + Λ + ( β ) (cid:0) Λ ′′ + ( β ) − Λ ′′ − ( β ) (cid:1) (Λ + ( β ) − Λ − ( β )) + 2Λ + ( β ) (cid:0) Λ ′ + ( β ) − Λ ′ − ( β ) (cid:1) (Λ + ( β ) − Λ − ( β )) β − " Λ ′′ − ( α )Λ + ( α ) − Λ − ( α ) − ′ − ( α ) (cid:0) Λ ′ + ( α ) − Λ ′ − ( α ) (cid:1) (Λ + ( α ) − Λ − ( α )) − Λ + ( α ) (cid:0) Λ ′′ + ( α ) − Λ ′′ − ( α ) (cid:1) (Λ + ( α ) − Λ − ( α )) + 2Λ − ( α ) (cid:0) Λ ′ + ( α ) − Λ ′− ( α ) (cid:1) (Λ + ( α ) − Λ − ( α )) α (4.1)and e t = Z βα + ( ζ ) − Λ − ( ζ ) dζ − Z β α + ( ζ ) − Λ − ( ζ ) dζ = Z α α + ( ζ ) − Λ − ( ζ ) dζ + Z ββ + ( ζ ) − Λ − ( ζ ) dζ = e β Λ + ( β ) − Λ − ( β ) −
12 Λ ′ + ( β ) − Λ ′ − ( β )(Λ + ( β ) − Λ − ( β )) e β + 16 Λ ′′ − ( β ) − Λ ′′ + ( β )(Λ + ( β ) − Λ − ( β )) + 2(Λ ′ + ( β ) − Λ ′ − ( β )) (Λ + ( β ) − Λ − ( β )) ! e β − e α Λ + ( α ) − Λ − ( α ) −
12 Λ ′ + ( α ) − Λ ′ − ( α )(Λ + ( α ) − Λ − ( α )) e α ! − Λ ′′ − ( α ) − Λ ′′ + ( α )(Λ + ( α ) − Λ − ( α )) + 2(Λ ′ + ( α ) − Λ ′ − ( α )) (Λ + ( α ) − Λ − ( α )) ! e α . (4.2) o prove Theorem 4.1, we need the following lemmas. Lemma 4.1.
Under the assumptions (H1)-(H5), it holds that e α = −
12 ( C e t + C e x ) +
14 ( C e t + C e x ) + C e t ! + −
12 ( C e t + C e x ) −
14 ( C e t + C e x ) + C e t ! , (4.3) where C i ( i = 1 , , ) depend only on the values of initial data at ( α , β ) .Proof. Under the assumptions (H1)-(H5), it holds thatΛ − ( α ) = Λ + ( β ) = 0 . (4 . f ′ ( α ) = Λ ′′ − ( α )Λ ( α ) − Λ − ( β )Λ ′′ + ( β ) − Λ − ( β )Λ ( α )+ Λ ′ + ( β )Λ − ( β ) (cid:2) Λ ′ + ( β ) − Λ ′ − ( β ) − (cid:0) Λ ′ + ( α ) − Λ ′ − ( α ) (cid:1)(cid:3) − Λ − ( β )Λ ( α ) < . (4.5)So Λ ′′ − ( α )Λ ( α ) − Λ − ( β )Λ ′′ + ( β )+Λ ′ + ( β )Λ − ( β ) (cid:2) Λ ′ + ( β ) − Λ ′ − ( β ) − (cid:0) Λ ′ + ( α ) − Λ ′ − ( α ) (cid:1)(cid:3) < . (4 . t and ˜ x up to third order of ˜ α and ˜ β to get an optimalestimate in (4.1) and (4.2). Noting the assumption (3.10) and using (4.1) and (4.2) leadto e t = − e β Λ − ( β ) −
12 Λ ′ + ( β ) − Λ ′ − ( β )Λ − ( β ) e β + 16 Λ ′′ − ( β ) − Λ ′′ + ( β )Λ − ( β ) − (cid:0) Λ ′ + ( β ) − Λ ′ − ( β ) (cid:1) Λ − ( β ) ! e β − e α Λ + ( α ) + 12 Λ ′ + ( α ) − Λ ′ − ( α )Λ ( α ) e α − Λ ′′ − ( α ) − Λ ′′ + ( α )Λ ( α ) + 2(Λ ′ + ( α ) − Λ ′ − ( α )) Λ ( α ) ! e α (4.7) nd e x = − Λ ′ + ( β )2Λ − ( β ) e β + 16 Λ ′′ + ( β ) − Λ − ( β ) − ′ + ( β ) (cid:0) Λ ′ + ( β ) − Λ ′ − ( β ) (cid:1) Λ − ( β ) ! e β − Λ ′ − ( α )2Λ + ( α ) e α − Λ ′′ − ( α )Λ + ( α ) − ′ − ( α ) (cid:0) Λ ′ + ( α ) − Λ ′ − ( α ) (cid:1) Λ ( α ) ! e α . (4.8)Noting (4.7), (4.8) and using the iterative method, we can obtain e x = B e α + B e t e α + B e t , (4 . B = − Λ ′ + ( β )Λ − ( β )2 < , B = Λ ′ − ( α ) < . B = − Λ ′′ − ( α )Λ ( α ) − Λ − ( β )Λ ′′ + ( β )6Λ ( α )+ Λ ′ + ( β )Λ − ( β )[Λ ′ + ( β ) − Λ ′ − ( β ) − (Λ ′ + ( α ) − Λ ′ − ( α ))]6Λ ( α ) > . (4.11)Solving equation (4.9) gives e α = −
12 ( C e t + C e x ) +
14 ( C e t + C e x ) + C e t ! + −
12 ( C e t + C e x ) −
14 ( C e t + C e x ) + C e t ! , where C = B B < , C = B B < , C = − B < . By (4.10) and (4.11), we observe that C i ( i = 1 , ,
3) depend only on the value of initialdata at ( α , β ). Thus, the lemma is proved. (cid:3) Lemma 4.2.
Under the assumptions (H1)-(H5), it holds that e α = − C C e t − C e xC e t , e x = o ( | e t | ) , ( − C e x ) , | e t | = o ( e x ) ,C ( − ( C e x )) , e x = O (1) | e t | (4 . for e t and e x sufficiently small, C stands for a constant and C i ( i = 1 , , are determinedby Lemma 4.1. roof. By Lemma 4.1, for simplicity, we may rewrite (4 .
3) as e α = (cid:16) A + ( A + B ) (cid:17) + (cid:16) A − ( A + B ) (cid:17) , (4 . A = −
12 ( C e t + C e x ) and B = C e t . We next prove Lemma 4.2 by distinguishing three cases:
Case I : A = o ( B ), i.e., e x = o ( | e t | ).By Taylor expansion, we have e α = " A + B (cid:18) A B (cid:19) + " A − B (cid:18) A B (cid:19) = (cid:20) A + B (cid:18) A B + o ( A B ) (cid:19)(cid:21) + (cid:20) A − B (cid:18) A B + o ( A B ) (cid:19)(cid:21) = B ((cid:20) AB + (1 + A B + o ( A B )) (cid:21) − (cid:20) − AB + (1 + A B + o ( A B )) (cid:21) ) = B (cid:20) A B + A B − (1 + A B − A B ) + o ( A B ) (cid:21) = B ( 2 A B + o ( A B )) = 2 AB − o ( A B − )= − C C e t − C e xC e t + o (˜ t + ˜ x ˜ t − + ˜ x ˜ t − ) . The special case − C C e t − C e xC e t = 0 implies that e α = o (˜ t ), which does not affect the mainresults of the paper, so we do not distinguish this special case anymore. Case II : B = o ( A ), i.e., | e t | = o ( e x ).By Taylor expansion, we have e α = " A + | A | (cid:18) BA (cid:19) + " A − | A | (cid:18) BA (cid:19) = (cid:20) A + | A | (cid:18) B A + o ( BA ) (cid:19)(cid:21) + (cid:20) A − | A | (cid:18) B A + o ( BA ) (cid:19)(cid:21) = | A | ((cid:20) sign ( A ) + (cid:18) B A + o ( BA ) (cid:19)(cid:21) + (cid:20) sign ( A ) − (cid:18) B A + o ( BA ) (cid:19)(cid:21) ) = ((2 + o (1)) A ) = ( − C e t − C e x ) = ( − C e x ) (1 + C ˜ t C ˜ x ) = ( − C e x ) (1 + C ˜ t C ˜ x + o ( ˜ t ˜ x )) = ( − C e x ) + o (˜ x ) . ase III : B = O (1)( A ), i.e., e x = O (1) | e t | .By Taylor expansion, we have e α = (cid:20) A + (cid:16) A + O (1) A (cid:17) (cid:21) + (cid:20) A − (cid:16) A + O (1) A (cid:17) (cid:21) = (cid:20) A + | A | (cid:16) O (1) (cid:17) (cid:21) + (cid:20) A − | A | (cid:16) O (1) (cid:17) (cid:21) = ((cid:20) (cid:16) O (1) (cid:17) (cid:21) + (cid:20) − (cid:16) O (1) (cid:17) (cid:21) ) A = ((cid:20) (cid:16) O (1) (cid:17) (cid:21) + (cid:20) − (cid:16) O (1) (cid:17) (cid:21) ) (cid:20) −
12 ( C e t + C e x ) (cid:21) , C (cid:18) −
12 ( C e t + C e x ) (cid:19) = C (cid:18) − C e x (cid:19) + o (˜ x ) , where C = (cid:18) (cid:16) O (1) (cid:17) (cid:19) + (cid:18) − (cid:16) O (1) (cid:17) (cid:19) , O (1) = 4 C sign ( e t )27 C ( O (1)) . (4 . (cid:3) Lemma 4.3.
Under the assumptions (H1)-(H5) and (4.7), it holds that Λ + ( β ) − Λ − ( α ) = D e t, e x = o ( | e t | ) ,D e x , | e t | = o ( e x ) ,D e t, e x = O (1) | e t | , (4 . where D i ( i = 1 , , depend only on the initial data at ( α , β ) .Proof. Iterating (4.7) two times and retain ˜ α to second order term, we obtain˜ β = − Λ − ( β ) " ˜ t + ˜ α Λ + ( α ) + Λ ′ + ( β ) − Λ ′ − ( β ) − (Λ ′ + ( α ) − Λ ′ − ( α ))2Λ ( α ) ˜ α . Then by assumptions (H1)-(H5) and above discussions, we haveΛ + ( β ) − Λ − ( α )= Λ + ( β ) − Λ + ( β ) + Λ − ( α ) − Λ − ( α )= Λ ′ + ( β ) e β − Λ ′ − ( α ) e α + 12 Λ ′′ + ( β ) e β −
12 Λ ′′ − ( α ) e α = − Λ ′ + ( β )Λ − ( β ) e t + M e α , (4.16) here M = Λ ′′ + ( β )Λ − ( β ) − Λ ′′ − ( α )Λ ( α ) − Λ ′ + ( β )Λ − ( β ) (cid:2) Λ ′ + ( β ) − Λ ′ − ( β ) − (Λ ′ + ( α ) − Λ ′ − ( α )) (cid:3) ( α ) . By (3.11), it holds that M = 0, thus, by Lemma 4.2, we have Case I : e x = o ( | e t | ).Λ + ( β ) − Λ − ( α ) = − Λ ′ + ( β )Λ − ( β ) e t + M (cid:18) − C C e t − C e xC e t (cid:19) = − Λ ′ + ( β )Λ − ( β ) e t + M C e x C e t = − Λ ′ + ( β )Λ − ( β ) e t , D e t, (4.17)where D = − Λ ′ + ( β )Λ − ( β ) . Case II : | e t | = o ( e x ).Λ + ( β ) − Λ − ( α ) = − Λ ′ + ( β )Λ − ( β ) e t + M ( − C e x ) = M ( − C e x ) , D e x , (4.18)where D = M C . Case III : e x = O (1) | e t | .Λ + ( β ) − Λ − ( α ) = − Λ ′ + ( β )Λ − ( β ) e t + M C (cid:18) −
12 ( C e x ) (cid:19) , D e t, (4.19)where D = − Λ ′ + ( β )Λ − ( β ) + M C (cid:18) C O (1)2 (cid:19) sign ( e t ) . Thus, the lemma is proved. (cid:3)
Proof of Theorem 4.1 .By (2.4), we have u ( t, x ) = λ + ( t, x ) + λ − ( t, x )2 = Λ + ( β ) + Λ − ( α )2 (4 . ρ ( t, x ) = 2 µλ + ( t, x ) − λ − ( t, x ) = 2 µ Λ + ( β ) − Λ − ( α ) . (4 . α, β ) (resp. ( α , β )) to denote the characteristic parametersdefined by (2.10) and (2.11) corresponding to ( t, x ) (resp. ( t , x )). n order to estimate u ( t, x ) at the blowup point ( t , x ), we firstly estimateΛ + ( β ) − Λ + ( β ) + Λ − ( α ) − Λ − ( α ) . (4 . + ( β ) − Λ + ( β ) + Λ − ( α ) − Λ − ( α )= Λ ′ + ( β ) e β + Λ ′ − ( α ) e α = Λ ′ + ( β ) (cid:18) − Λ − ( β ) e t − Λ − ( β )Λ + ( α ) e α (cid:19) + Λ ′ − ( α ) e α = − Λ ′ + ( β )Λ − ( β ) e t + − Λ ′ + ( β )Λ − ( β ) + Λ ′ − ( α )Λ + ( α )Λ + ( α ) e α = − Λ ′ + ( β )Λ − ( β ) e t − ′ + ( β )Λ − ( β )Λ + ( α ) e α. (4.23)By Lemma 4.2, we have Case I : e x = o ( | e t | ) . We haveΛ + ( β ) + Λ − ( α )= − Λ ′ + ( β )Λ − ( β ) e t − ′ + ( β )Λ − ( β )Λ + ( α ) (cid:18) − C C e t − C e xC e t (cid:19) = − Λ ′ + ( β )Λ − ( β ) + 2Λ ′ + ( β )Λ − ( β ) C Λ + ( α ) C ! e t + 2Λ ′ + ( β )Λ − ( β ) C Λ + ( α ) C e x e t , C e t + C e x e t = C e x e t + o (˜ t ) , (4.24)where C = − Λ ′ + ( β )Λ − ( β ) + 2Λ ′ + ( β )Λ − ( β ) C Λ + ( α ) C = 0 , C = 2Λ ′ + ( β )Λ − ( β ) C Λ + ( α ) C . Case II : | e t | = o ( e x ) . In this case, we getΛ + ( β ) + Λ − ( α ) = − Λ ′ + ( β )Λ − ( β ) e t − ′ + ( β )Λ − ( β )Λ + ( α ) ( − C e x ) = − ′ + ( β )Λ − ( β )Λ + ( α ) ( − C e x ) , C e x , (4.25)where C = 2Λ ′ + ( β )Λ − ( β )Λ + ( α ) C . Case III : e x = O (1) | e t | . e obtainΛ + ( β ) + Λ − ( α ) = − Λ ′ + ( β )Λ − ( β ) e t − ′ + ( β )Λ − ( β )Λ + ( α ) C (cid:18) − C e x (cid:19) = − ′ + ( β )Λ − ( β )Λ + ( α ) C (cid:18) − C e x (cid:19) , C e x , (4.26)where C = 2Λ ′ + ( β )Λ − ( β )Λ + ( α ) C (cid:18) C (cid:19) . In order to estimate u x , ρ x , u t and ρ t , we have to estimate ( λ + ) x , ( λ − ) x , ( λ + ) t , ( λ − ) t .By (3.2), we can obtain the estimates on ( λ + ) x and ( λ − ) x .It follows from the system (2.5) that( λ + ) t = − λ − ( λ + ) x = − Λ − ( α )Λ ′ + ( β ) Λ + ( β ) − Λ − ( β )Λ + ( β ) − Λ − ( α )= − Λ ′ − ( α )Λ ′ + ( β ) Λ + ( β ) − Λ − ( β )Λ + ( β ) − Λ − ( α ) e α (4.29)and ( λ − ) t = − λ + ( λ − ) x = − Λ + ( β )Λ ′ − ( α ) Λ + ( α ) − Λ − ( α )Λ + ( β ) − Λ − ( α )= − Λ ′ + ( β )Λ ′ − ( α ) Λ + ( α ) − Λ − ( α )Λ + ( β ) − Λ − ( α ) e β = Λ − ( β )Λ ′ + ( β )Λ ′ − ( α ) Λ + ( α ) − Λ − ( α )Λ + ( β ) − Λ − ( α ) (cid:18)e t + e α Λ + ( α ) (cid:19) . (4.30)We now estimate u x , u t , ρ x and ρ t For u x , noting (4.20)-(4.21) and (3.2), by direct calculations, we have u x = ( λ + ) x + ( λ − ) x , M Λ + ( β ) − Λ − ( α ) , (4 . M = Λ ′ + ( β )(Λ + ( β ) − Λ − ( β ))2 + Λ ′ − ( α )(Λ + ( α ) − Λ − ( α ))2 . Similarly, for u t , by (4.29) and (4.30), we have u t = ( λ + ) t + ( λ − ) t , M e t + M e α Λ + ( β ) − Λ − ( α ) , (4 . M = Λ ′ + ( β )Λ ′ − ( α )Λ − ( β )(Λ + ( α ) − Λ − ( α ))and M = Λ ′ + ( β )Λ ′ − ( α )Λ − ( β ) (Λ + ( α ) − Λ − ( α ))Λ + ( α ) − Λ ′ − ( α )Λ ′ + ( β ) (Λ + ( β ) − Λ − ( β )) . or ρ x , we have ρ x = − µ (( λ + ) x − ( λ − ) x )( λ + − λ − ) , M (Λ + ( β ) − Λ − ( α )) , (4 . M = 2 µ (cid:16) Λ ′ + ( β ) (Λ + ( β ) − Λ − ( β )) − Λ ′ − ( α ) (Λ + ( α ) − Λ − ( α )) (cid:17) . For ρ t , we have ρ t = − µ (( λ + ) t − ( λ − ) t )( λ + − λ − ) , M e t + M e α (Λ + ( β ) − Λ − ( α )) (4 . M = 2 µ Λ ′ + ( β )Λ ′ − ( α )Λ − ( β ) (Λ + ( α ) − Λ − ( α ))and M = 2 µ " Λ ′ + ( β )Λ ′ − ( α )Λ − ( β ) (Λ + ( α ) − Λ − ( α ))Λ + ( α ) + Λ ′ − ( α )Λ ′ + ( β ) (Λ + ( β ) − Λ − ( β )) . Then, by (4.20)-(4.21) and (4.31)-(4.34), we have
Case I : e x = o ( | e t | ) . It holds that | u ( t, x ) − u ( t , x ) | ≤ | C || e t | + | C | (cid:12)(cid:12)(cid:12)(cid:12) e x e t (cid:12)(cid:12)(cid:12)(cid:12) , F | e t | + F (cid:12)(cid:12)(cid:12)(cid:12) e x e t (cid:12)(cid:12)(cid:12)(cid:12) , (4 . | u x | ≤ | M Λ + ( β ) − Λ − ( α ) | ≤ | M | | − Λ ′ + ( β )Λ − ( β ) e t | = 4 | M | Λ ′ + ( β )Λ − ( β ) | e t | − , F | e t | − , (4 . | u t | ≤ | M || e t | + | M || e α || D e t | , F + F | e x || e t | , (4 . | ρ | ≤ µ Λ ′ + ( β )Λ − ( β ) | e t | − , F | e t | − , (4 . | ρ x | ≤ | M | (cid:0) Λ ′ + ( β )Λ − ( β ) (cid:1) | e t | − , F | e t | − , (4 . | ρ t | ≤ | M || e t | + | M || e α || D e t | , F | e t | + F | e x || e t | . (4 . Case II : | e t | = o ( e x ) . We have | u ( t, x ) − u ( t , x ) | ≤ | C || e x | , F | e x | , (4 . | u x | ≤ | M || D ˜ x | , F | e x | − , (4 . u t | ≤ | M e t | + | M e α | ( D e x ) , F | e x | − , (4 . | ρ | ≤ µD ( e x ) − , F | e x | − , (4 . | ρ x | ≤ | M | D | e x | − , F | e x | − , (4 . | ρ t | ≤ | M e t | + | M e α | ( D e x ) , F | e x | − . (4 . Case III : e x = O (1) | e t | . We obtain | u ( t, x ) − u ( t , x ) | ≤ | C || e x | , F | e x | , (4 . | u x | ≤ F | e t | − , (4 . | u t | ≤ | M e t | + | M C ( C O (1)) || e t | | D e t | , N + N | e t | − = F | e t | − , (4 . | ρ | ≤ F | e t | − , (4 . | ρ x | ≤ F | e t | − , (4 . | ρ t | ≤ | M e t | + | M C ( C O (1)) || e t | | D e t | , N | e t | − + N | e t | − = F | e t | − . (4 . Remark 4.2.
From the above discussions, it is easy to say that the constants derived inabove estimates are not equal to zero for initial data satisfying assumptions (H1)-(H5). analysis of singularity In this section we shall construct physical solutions with new kind of singularity forthe system (2 . u t + f ( u ) x = 0 , t > , x ∈ R , (5 . u (0 , x ) = u ( x ) , x ∈ R , where u ( t, x ) = ( u , · · · , u n )( t, x ) ∈ R n , n ≥ f ( u ) is the flux vector-valued functionin some open set Ω ⊂ R n . efinition 5.1. A bounded measurable function u ( t, x ) is called a weak solution of theCauchy problem (5.1) with bounded and measurable initial data u , provided that Z Z t ≥ ( uφ t + f ( u ) φ x ) d x d t + Z t =0 u φdx = 0 (5 . holds for all φ ∈ C ( R + × R ) , where C denotes the class of C functions φ , which vanishoutside a compact subset in t ≥ . In this paper, we generalize the above definition as follows
Definition 5.2.
A measurable function u ( t, x ) is called a weak solution of the Cauchyproblem (5.1) with bounded and measurable initial data u , provided that (5.2) holds forall φ ∈ C ( R + × R ) . Corollary 5.1. If u is a weak solution, then it holds that: lim ǫ → Z Z ( t,x ) ∈ D ǫ ( uφ t + f ( u ) φ x ) d x d t = 0 , (5 . where D ǫ = { ( t, x ) | | t − t | ≤ ǫ, | x − x | ≤ ǫ } and ( t , x ) is a blowup point.Proof. If (5.2) holds, then taking φ ∈ C ( D ǫ ) gives Z Z ( t,x ) ∈ D ǫ ( uφ t + f ( u ) φ x ) dxdt = 0 . This is nothing but (5.3) (cid:3)
Let u ± = u ( t, x ( t ) ± , where x ( t ) is a smooth curve across which u has a jump discontinuity. As in the traditionalsense, we can also get the Rankine-Hugoniot condition (see [18]) s [ u ] = [ f ( u )] , (5 . s = dx ( t ) dt is the speed of discontinuity, [ u ] = u + − u − , the jump across x ( t ) andsimilarly, [ f ] = f ( u + ) − f ( u − ), at which we do not require that u has well-defined limits onboth sides of x = x ( t ), i.e., u may be infinity on either side of the discontinuity x = x ( t ). emark 5.1. In our definition, we do not require that u ( t, x ) is bounded everywhere,while, we need the singular integral in the left hand side of (5 . is convergent. Definition 5.3. u = u ( t, x ) of system (5.1) is said to be the “Delta-like solution”, if itsatisfies Definition 5.2 and u ( t, x ) is smooth except on some points or curves or otherdomains, on which u = ∞ . Remark 5.2.
By Definition 5.3, the “Delta-like” solution is different from the “shock-wave” solution. Here, the density of system (2.1) is unbounded.
Lemma 5.1.
For system (2.1), if x = x ( t ) is a curve of discontinuity and ρ has a jumpacross x ( t ) , define ρ on both sides of x = x ( t ) as ρ ± = ρ ( t, x ( t ) ± . Then dx ( t ) dt = u + = u − . (5 . where u ± = u ( t, x ( t ) ± are the right and left limits, respectively.Proof. If x = x ( t ) is a curve of discontinuity of system (2.1), then, by (5.4), we have s ( ρ + − ρ − ) = ρ + u + − ρ − u − (5 . s ( ρ + u + − ρ − u − ) = ρ + ( u + ) − ρ − ( u − ) . (5 . ρ + = ρ − . Then by (5.6) and (5.7), we have( ρ + − ρ − )( ρ + ( u + ) − ρ − ( u − ) ) = ( ρ + u + − ρ − u − ) . By a simple calculation, we get u + = u − , thus, by (5.6), we obtain dx ( t ) dt = u + = u − . (cid:3) Remark 5.3.
Since on both sides of the discontinuity x = x ( t ) , the pressure p = 0 , (5.7)holds accordingly. Theorem 5.1.
Under the assumptions (H1)-(H5) in Section 3, the solution of the Cauchyproblem (2.1), (2.6) constructed by the method of characteristics satisfy (5.3) in the strip { ( t, x ) | t ∈ [0 , t ] , x ∈ R } and t is defined by (3.10). roof. It suffices to check thatlim ǫ → Z Z ( t,x ) ∈ D − ǫ ( ρφ t + ( ρu ) φ x ) d x d t = 0 (5 . ǫ → Z Z ( t,x ) ∈ D − ǫ (( ρu ) φ t + ( ρu ) φ x ) d x d t = 0 , (5 . D − ǫ = { ( t, x ) | | x − x | ≤ ǫ, t − ǫ ≤ t ≤ t } By Lemma 4.1, we have C < . We prove Theorem 5.1 by distinguishing the following three possible cases: B = O (1) A , B = o ( A ) and A = o ( B ) , where A and B are defined in the proof of Lemma 4.2. From B = A , we have C e t
27 = 14 ( C e t + C e x ) . (5 . e t ≤ . Equation (5.10) defines two curves passing through ( t , x ) read e x = 2 C ( − C ) (27) C ( − e t ) , G ( − e t ) (5 . e x = − C ( − C ) (27) C ( − e t ) , − G ( − e t ) , (5 . D − ǫ into T and T defined by T = { ( t, x ) ∈ D − ǫ | B ≥ A } and T = { ( t, x ) ∈ D − ǫ | B < A } . Thus, it suffices to prove that (5.8) and (5.9) hold in T S T .Case A = o ( B ), namely, e x = o ( | e t | )By Theorem 4.1, we have the following asymptotic solutions: ρ ≈ k e t − , (5 . ≈ k e t + k e x e t , (5 . ρu ≈ m + m e x ˜ t − , (5 . ρu ≈ m e t + m e x e t − . (5 . B = O (1) A , namely, e x = O (1) | e t | . We obtain ρ ≈ k e t − = k e x − , (5 . u ≈ k e x , (5 . ρu ≈ m e x − , (5 . ρu ≈ m . (5 . B = o ( A ), namely, | e t | = o ( e x ).By Theorem 4.1, we have the following asymptotic solutions: ρ ≈ k e x − , (5 . u ≈ k e x , (5 . ρu ≈ m e x − , (5 . ρu ≈ m , (5 . k i ( i = 1 , · · · ,
8) and m i ( i = 1 , · · · ,
8) are constants depending only on the initialdata at ( α , β ). By (5.8) and (5.9), it follows that P = lim ǫ → Z Z ( t,x ) ∈ D − ǫ ( ρφ t + ( ρu ) φ x ) d x d t = lim ǫ → Z Z ( t,x ) ∈ T ( ρφ t + ( ρu ) φ x ) d x d t + lim ǫ → Z Z ( t,x ) ∈ T ( ρφ t + ( ρu ) φ x ) d x d t , P + P (5.25) nd Q = lim ǫ → Z Z ( t,x ) ∈ D ǫ (( ρu ) φ t + ( ρu ) φ x ) d x d t = lim ǫ → Z Z ( t,x ) ∈ T (( ρu ) φ t + ( ρu + p ) φ x ) d x d t + lim ǫ → Z Z ( t,x ) ∈ T (( ρu ) φ t + ( ρu + p ) φ x ) d x d t , Q + Q . (5.26)We next prove | P i | = 0 ( i = 1 ,
2) and | Q i | = 0 ( i = 1 , . Define K = { ( t, x ) | e x = o ( | e t | ) } ,K = { ( t, x ) | e x = O (1) | e t | } and K = { ( t, x ) | | e t | = o ( e x ) } . Then by (5.13)-(5.24), we obtain | P | ≤ lim ǫ → Z Z ( t,x ) ∈ T ( | ρ || φ t | + | ( ρu ) || φ x | ) d x d t ≤ max | φ t | lim ǫ → Z Z ( t,x ) ∈ T | ρ | d x d t + max | φ x | lim ǫ → Z Z ( t,x ) ∈ T | ρu | d x d t ≤ max | φ t | lim ǫ → Z Z ( t,x ) ∈ T T K | ρ | d x d t + max | φ x | lim ǫ → Z Z ( t,x ) ∈ T T K | ρu | d x d t + max | φ t | lim ǫ → Z Z ( t,x ) ∈ T T K | ρ | d x d t + max | φ x | lim ǫ → Z Z ( t,x ) ∈ T T K | ρu | d x d t ≤ max | φ t | lim ǫ → Z Z ( t,x ) ∈ T T K | k e t − | d x d t + max | φ x | lim ǫ → Z Z ( t,x ) ∈ T T K | m + m e x ˜ t − | d x d t + max | φ t | lim ǫ → Z Z ( t,x ) ∈ T T K | k e x − | d x d t + max | φ x | lim ǫ → Z Z ( t,x ) ∈ T T K | m e x − | d x d t ≤ max | φ t | lim ǫ → Z G ( − e τ ) − G ( − e τ ) Z − ǫ ( | k e x − | + | k e x − | ) d e td e x + max | φ x | lim ǫ → Z G ( − e τ ) − G ( − e τ ) Z − ǫ ( | m + m e x ˜ t − | + | m e x − | ) d e td e x . | P | ≤ lim ǫ → Z Z ( t,x ) ∈ T ( | ρ || φ t | + | ( ρu ) || φ x | ) d x d t ≤ max | φ t | lim ǫ → Z Z ( t,x ) ∈ T | ρ | d x d t + max | φ x | lim ǫ → Z Z ( t,x ) ∈ T | ρu | d x d t ≤ max | φ t | lim ǫ → Z Z ( t,x ) ∈ T T K | ρ | d x d t + max | φ x | lim ǫ → Z Z ( t,x ) ∈ T T K | ρu | d x d t + max | φ t | lim ǫ → Z Z ( t,x ) ∈ T T K | ρ | d x d t + max | φ x | lim ǫ → Z Z ( t,x ) ∈ T T K | ρu | d x d t ≤ max | φ t | lim ǫ → Z Z ( t,x ) ∈ T T K | k e x − | d x d t + max | φ x | lim ǫ → Z Z ( t,x ) ∈ T T K | m e x − | d x d t + max | φ t | lim ǫ → Z Z ( t,x ) ∈ T T K | k e x − | d x d t + max | φ x | lim ǫ → Z Z ( t,x ) ∈ T T K | m e x − | d x d t ≤ max | φ t | lim ǫ → ( Z − G ( − e τ ) − ǫ Z − ǫ + Z ǫG ( − e τ ) Z − ǫ )( | k e x − | + | k e x − | ) d e td e x + max | φ x | lim ǫ → ( Z − G ( − e τ ) − ǫ Z − ǫ + Z ǫG ( − e τ ) Z − ǫ )( | m e x − | + | m e x − | ) d e td e x = 0 . Here max | φ x | = sup {| φ x ( t, x ) | | ( t, x ) ∈ D − ǫ } , max | φ t | = sup {| φ t ( t, x ) | | ( t, x ) ∈ D − ǫ } and ˜ τ is defined by (5.11) and (5.12). So we have | P | →
0, as ǫ →
0. That is to say, (5.8)holds.Similarly, we can prove (5.9). Thus, the theorem is proved. (cid:3)
Remark 5.4.
Here and throughout the following, we will use the convention ¯ A ≈ ¯ B whenever ¯ C − ¯ A ≤ ¯ B ≤ ¯ C ¯ A for a constant ¯ C = 0 . Delta-like solutions
In this section, by the method of characteristics, we construct some weak solutionswith a new kind of singularities, named “Delta-like” solution. e first consider a simple case, in which we assumeAssumption (A1): Λ − ( x ) < Λ + ( x ) , ∀ x ∈ R ; (6 . α and β with α < β satisfyingAssumption (A2): Λ − ( α ) = Λ + ( β ) = 0; (6 . ′ − ( α ) = Λ ′ + ( β ) = 0; (6 . ′′ − ( α ) < , Λ ′′ + ( β ) > . (6 . ± ( x ) satisfying assumptions (A1)-(A4) are shown in Figure 6. ✲✻ t xα β Λ − ( x ) Λ + ( x ) Figure 6. Λ ± ( x ) satisfying the assumptions (A1)-(A4)We now derive the Delta-like solution with point-shape singularity Theorem 6.1.
Under the assumptions (A1)-(A4), it holds that in the neighborhood ofthe blowup point | ρ | ≤ B | e t | − , e x = o ( e t ) ,B | e x | − , e t = o ( e x ) ,B | e x | − , e x = O (1) e t and | u | ≤ B | e t | , e x = o ( e t ) ,B | e x | , e t = o ( e x ) ,B | e x | , e x = O (1) e t for sufficiently small e t and e x defined in Section 4 and B i ( i = 1 , · · · , are positiveconstants depending only on the initial data at ( α , β ) . Furthermore, the solution ( ρ, u ) is Delta-like solution, which we call it Delta-like solution with point-shape singularity. Before proving Theorem 6.1, we need the following lemmas emma 6.1. Under the assumptions (A1)-(A4), there is only one singular point, i.e., ( t , x ) defined by (3.12) and (3.13), on which ρ = ∞ and away from ( t , x ) , ρ ( t, x ) isfinite.Proof. By (3.12), (3.13), (4.21) and (6.2), we observe that at ( t , x ), ρ = ∞ .Suppose that there exists another point ( t, x ) = ( t , x ) such that ρ ( t, x ) = ∞ . from( t, x ), there exist only two characteristics intersecting the x -axis at α and β respectively.By (4.21) we have Λ − ( α ) = Λ + ( β ) . (6 . α = α , then by the assumptions (A1)-(A4) we haveΛ − ( α ) < , Λ + ( β ) ≥ , ∀ β ∈ R . This contradicts to (6.15).Similarly, it is easy to show that the assumption β = β also leads to a contradiction.Thus, the lemma is proved. (cid:3) Under the assumptions (A1)-(A4), the characteristics can be depicted as follows:the characteristics x − and x + passing through (0 , α ) and (0 , β ) respectively tangentat ( t , x ) and then they turn away from each other (see Figure 7). ✲✻ t xα β x − x − x + x + ( t , x ) Figure 7.
The characteristics under assumptions (A1)-(A4).By the same method as Lemmas 4.1-4.3, we get the following two lemmas withoutproof.
Lemma 6.2.
Under the assumptions (A1)-(A4), in the neighborhood of ( t , x ) , it holdsthat e β =
12 ( − B e x − B e t ) + r
14 ( − B e x − B e t ) + 127 ( B e t ) !
12 ( − B e x − B e t ) − r
14 ( − B e x − B e t ) + 127 ( B e t ) ! − B e t , where B i ( i = 7 , · · · , are constants depending only on the initial data at ( α , β ) . Lemma 6.3.
Under the assumptions (A1)-(A4), in the neighborhood of ( t , x ) , it holdsthat e β = B e t, e x = o ( e t ) ,B e x , e t = o ( e x ) ,B e x , e x = O (1) e t , Λ + ( β ) − Λ − ( α ) = B e t , e x = o ( e t ) ,B e x , e t = o ( e x ) ,B e x , e x = O (1) e t and Λ + ( β ) + Λ − ( α ) = B e t , e x = o ( e t ) ,B e x , e t = o ( e x ) ,B e x , e x = O (1) e t where B i ( i = 11 , · · · , are constants depending only on the initial data at ( α , β ) . The proof of Theorem 6.1.
The behavior of ρ and u can be derived easily fromthe above two lemmas. By the same method as Theorem 5.1, in the integral domain D ǫ defined below, we have | ρ | ≤ | e x | − and the orders of | ρu | and | ρu | are higher than theorder of | ρ | , thus, (5.3) holds obviously. Next, we prove that system (2.1) satisfies theDefinition 5.2. First we consider the first equation of (2.1).Define D ǫ = { ( t, x ) | | t − t | ≤ ǫ, | x − x | ≤ ǫ } . For arbitrary φ ( t, x ) ∈ C , it holds that Z Z t ≥ ( ρφ t + ρuφ x ) dxdt + Z t =0 ρ φdx = Z Z { t ≥ }− D ǫ ( ρφ t + ρuφ x ) dxdt + Z t =0 ρ φ + Z Z D ǫ ( ρφ t + ρuφ x ) dxdt = (cid:18)Z ∞ Z x − ǫ −∞ + Z ∞ Z + ∞ x + ǫ + Z t − ǫ Z x + ǫx − ǫ + Z ∞ t + ǫ Z x + ǫx − ǫ (cid:19) ( ρφ t + ρuφ x ) dxdt + Z Z D ǫ ( ρφ t + ρuφ x ) dxdt + Z t =0 ρ φdx = (cid:18)Z ∞ Z x − ǫ −∞ + Z ∞ Z + ∞ x + ǫ + Z t − ǫ Z x + ǫx − ǫ + Z ∞ t + ǫ Z x + ǫx − ǫ (cid:19) ( ρ t φ + ( ρu ) x φ ) dxdt Z + ∞−∞ ρ φdx + Z x + ǫx − ǫ ( ρ ( t − ǫ ) φ ( t − ǫ ) − ρ ( t + ǫ ) φ ( t + ǫ )) dx + Z Z D ǫ ( ρφ t + ρuφ x ) dxdt + Z t + ǫt − ǫ ( ρuφ ( x − ǫ ) − ρuφ ( x + ǫ )) dt + Z + ∞−∞ ρ φdx = (cid:18)Z ∞ Z x − ǫ −∞ + Z ∞ Z + ∞ x + ǫ + Z t − ǫ Z x + ǫx − ǫ + Z ∞ t + ǫ Z x + ǫx − ǫ (cid:19) ( ρ t + ( ρu ) x ) φdxdt + Z x + ǫx − ǫ ( ρφ ( t − ǫ ) − ρφ ( t + ǫ )) dx + Z t + ǫt − ǫ ( ρuφ ( t − ǫ ) − ρuφ ( t + ǫ )) dt + Z Z D ǫ ( ρφ t + ρuφ x ) dxdt , M + M + M + M . In the limit ǫ → M → M → M , we have | M | ≤ max | φ | Z x + ǫx − ǫ | ρ | dx ≤ max | φ | Z x + ǫx − ǫ | x − x | − dx ≤ max | φ | ǫ , where max | φ | = sup {| φ ( t, x ) | | ( t, x ) ∈ D ǫ } . So M tends to zero. For M , since | ρuφ | is finite on ( t − ǫ, t + ǫ ). Thus, M → ǫ →
0. Similarly, for the second equation of (2.1) we have
Z Z t ≥ ( ρuφ t + ( ρu + p ) φ x ) dxdt + Z t =0 ρ u φdx = 0 . In this subsection, we are going to investigate another important Delta-like solutionnamed Delta-like solution with line-shape singularity. To do so, we assume thatAssumption ( B − ( x ) < Λ + ( x ) , ∀ x ∈ R ; (6 . α and β with α < β satisfyingAssumption ( B − ( α ) = Λ + ( β ) = 0 , ∀ β ≤ β ≤ b β ; (6 . − ( α ) = Λ + ( β ) = 0 , ∀ b α ≤ α ≤ α ; ) ssumption ( B ′ − ( α ) = Λ ′ + ( β ) = 0; (6 . B ′′ − ( α ) < , Λ ′′ + ( β ) = 0; (6 . . Λ ′′ − ( α ) = 0 , Λ ′′ + ( β ) > . )Λ ± ( x ) satisfying assumptions ( B B
4) are shown in Figure 8. ✲✻ ✲✻ t xα β Λ − ( x ) Λ + ( x ) 0 t xα β Λ − ( x ) Λ + ( x ) Figure 8. Λ ± ( x ) satisfying the assumptions ( B B b t = Z b βα + ( ζ ) − Λ − ( ζ ) dζ . Under the assumptions ( B B Theorem 6.2.
Under the assumptions ( B )-( B ), it holds that lim ǫ → Z b t + ǫt − ǫ Z x + ǫx − ǫ ( ρφ t + ρuφ x ) dxdt = 0 (6 . and lim ǫ → Z b t + ǫt − ǫ Z x + ǫx − ǫ ( ρuφ t + ( ρu + p ) φ x ) dxdt = 0 . (6 . Furthermore, the solution derived by the method of characteristics is a Delta-like solutionand we call it Delta-like solution with line-shape singularity.
Before proving Theorem 6.2, we need the following lemmas.
Lemma 6.4.
Under the assumptions ( B )-( B ), the singularities form a line L , { ( t, x ) | x = x , t ≤ t ≤ b t } , and on this line, it holds that ρ = ∞ , while off the line, ρ is finiteand smooth, we denote this set by L c . roof. By (4.21), ρ = ∞ if and only if Λ + ( β ) = Λ − ( α ) , on the other hand, by (2.10)-(2.11),(3.12)-(3.13) and the assumptions ( B B β ≤ β ≤ b β , x = 12 (cid:26) α + β + Z βα Λ + ( ζ ) + Λ − ( ζ )Λ + ( ζ ) − Λ − ( ζ ) dζ (cid:27) = 12 (cid:26) α + β + β − β + Z β α Λ + ( ζ ) + Λ − ( ζ )Λ + ( ζ ) − Λ − ( ζ ) dζ + Z ββ Λ + ( ζ ) + Λ − ( ζ )Λ + ( ζ ) − Λ − ( ζ ) dζ (cid:27) = 12 (cid:26) x + β − β + Z ββ ( − dζ (cid:27) = x and t = Z βα + ( ζ ) − Λ − ( ζ ) dζ = Z β α + ( ζ ) − Λ − ( ζ ) dζ + Z ββ + ( ζ ) − Λ − ( ζ ) dζ = t + Z ββ + ( ζ ) − Λ − ( ζ ) dζ ≥ t . Moreover, since β ≤ b β , we have t ≤ b t . So, the mapping Π maps the curve Λ − ( α ) = Λ + ( β )into L and ( t , x ) is the blowup point. Passing through any point ( t, x ) ∈ L c , there existonly two characteristics which intersect the x -axis at α and β with α < β and satisfyingΛ − ( α ) = Λ + ( β ) , i.e., ρ < ∞ . Thus, the proof of Lemma 6.4 is completed. (cid:3)
Remark 6.1.
The characteristics under the assumptions ( B )-( B ) can be depicted asfollows: the characteristics x − and x + passing through (0 , α ) and (0 , β ) , respectively, aretangent at ( t , x ) , and then they turn away from each other when t > t . L is the envelopeof the characteristics passing through the points (0 , α ) and (0 , β ) in which β ≤ β ≤ b β (see Figure 9). ✲✻ t x α α α β β β Lx − x + Figure 9.
A sketch of the characteristics under the assumptions ( B B emma 6.5. Under the assumptions ( B )-( B ), in the neighborhood of ( t, x ) ∈ L definedin Lemma 6.4, it holds that e α = V e x , Λ + ( β ) − Λ − ( α ) = V e x , Λ + ( β ) + Λ − ( α ) = − V e x . Where V i ( i = 1 , are constants depending only on the initial data at α .Proof. By the assumptions ( B B
4) and (4.2), we have e x = − ′′ − ( α ) − Λ ′′ + ( α )Λ + ( α ) ! e α . So e α = s − + ( α )2Λ ′′ − ( α ) − Λ ′′ + ( α ) e x , V e x . On the other hand Λ + ( β ) − Λ − ( α ) = − Λ ′′ − ( α )2 e α , V e x and Λ + ( β ) + Λ − ( α ) = Λ ′′ − ( α )2 e α , − V e x . Thus, the lemma is proved. (cid:3)
By the above two lemmas, we are ready to prove Theorem 6.2
The proof of Theorem 6.2 .By Lemma 6.5, | ρ | ≤ V | e x | − , | ρu | ≤ µ and | ρu | ≤ µV | e x | , in b D ( ǫ ) , where b D ( ǫ ) = { ( t, x ) | t − ǫ ≤ t ≤ b t + ǫ, | x − x | ≤ ǫ } . Obviously, the singular integral (5.3) in b D ( ǫ ) is convergent. By the same method asTheorem 6.1, for arbitrary φ ∈ C ( R + × R ), we have Z Z t ≥ ( ρφ t + ρuφ x ) dxdt + Z t =0 ρ φdx = Z b t + ǫt − ǫ Z x + ǫx − ǫ ( ρφ t + ρuφ x ) dxdt + Z b t + ǫt − ǫ ( ρuφ ( x + ǫ ) − ρuφ ( x − ǫ )) dt , N + N . hen ǫ →
0, by (5.3) we have N → . For N , since | ρu | ≤ µ , | b t − t | < ∞ andlim ǫ → ( ρu ( x − ǫ ) − ρu ( x + ǫ )) = ( − µ − ( − µ )) = 0 , by the dominated convergence theorem, N → . Similarly, when ǫ →
0, we have
Z Z t ≥ ( ρuφ t + ( ρu + p ) φ x ) dxdt + Z t =0 ρ u φdx → This section is a continuation of the previous subsection 6.2. The difference is theassumptions on initial data. Here we assumeAssumption ( C − ( x ) < Λ + ( x ) , ∀ x ∈ R ; (6 . α , β and α < β satisfyingAssumption ( C − ( α ) = Λ + ( β ) = 0 , ∀ b α ≤ α ≤ α , ∀ β ≤ β ≤ b β ; (6 . C ′ − ( α ) = Λ ′ + ( β ) = 0; (6 . C ′′ − ( α ) = 0 , Λ ′′ + ( β ) = 0 . (6 . ± ( x ) satisfying assumptions ( C C
4) can be shown in Figure 10. ✲✻ t xα β Λ − ( x ) Λ + ( x ) Figure 10.
A sketch of Λ ± ( x ) satisfying the assumptions ( C C y the same method as Lemma 6.4, we have the following lemma. Lemma 6.6.
Under the assumptions ( C )-( C ), the singularities form a line L which isdefined as Lemma 6.4. The density ρ is infinite on the line L , while is finite and smoothoff the line. Remark 6.2.
Under the assumptions ( C )-( C ), the characteristics can be shown inFigure 11. The difference between the assumptions ( B )-( B ) and the assumptions ( C )-( C ) is that once the characteristic touch at the line L , they will remain in contact. ✲✻ t x α α α β β β Lx − x + Figure 11.
A sketch of the characteristics under the assumptions ( C C Remark 6.3.
On the line L the solution constructed by the method of characteristicssatisfies u + = u − = 0 , i.e., it satisfies (5.5) under the assumptions ( B )-( B ) and ( C )-( C ). conclusion In this paper, we study the behavior of one-dimensional Chaplygin gas. In particular,we analyze the formation of singularities for such a system. We show that these singular-ities are very different from the traditional formation of singularities, such as in the caseof shock wave formation. We call these new type singularities “Delta-like” singularitiessince the densities become infinite at the singularities. Depending on the initial condi-tions, different types of Delta-like singularities can form, such as Delta-like solution withpoint-shape singularity and Delta-like solution with line-shape singularity (including TypeI and II). For convenience, we assume that the initial data only leads to the formation ofone Delta-like singularity. It is straight forward to generalize this to the cases which allowmany Delta-like singularities to form. More specially, we can generalize the assumptions(A2)-(A4) to the case: there exist numerous α i , β j ( i, j = 0 , , , · · · , n ) satisfying ssumptions (A2’): Λ − ( α i ) = Λ + ( β j ) = 0; (7 . ′ − ( α i ) = Λ ′ + ( β j ) = 0; (7 . ′′ − ( α i ) < , Λ ′′ + ( β j ) > . (7 . Acknowledgements.
Kong and Wei thank A. Bressan for helpful discus-sion. The work of Kong and Wei was supported in part by the NNSF of China (GrantNo.: 11271323), Zhejiang Provincial Natural Science Foundation of China (Grant No.:Z13A010002) and a National Science and Technology Project during the twelfth five-yearplan of China (2012BAI10B04). The work of Zhang was supported by the Research GrantCouncil of HKSAR (Grant No.: CityU 103509).
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